This thesis is devoted to proving the following:
For all (u1, u2, u3, u4) in a Zariski dense open subset of C4 there is a genus 3 curve X(u1, u2, u3, u4) with the following properties:
1. X(u1, u2, u3, u4) is not hyperelliptic.
2. End(Jac((X(u1, u2, u3, u4))) ⊗Q contains the real cubic field Q(ζ7+ζ7-1) where ζ7 is a primitive 7th root of unity.
3. These curves X(u1, u2, u3, u4) define a three-dimensional subvariety of the moduli space of genus 3 curves M3.
4. The curve X(u1, u2, u3, u4) is defined over the field Q(u1, u2, u3, u4), and the endomorphisms are defined over Q(ζ7, u1, u2, u3, u4).
This theorem is a joint result of J. W. Hoffman, Ryotaro Okazaki,Yukiko Sakai, Haohao Wang and Zhibin Liang. My contribution to this project is the following: (1) Verification of property 3 above. This is accomplished in two ways. One utilizes the Igusa invariants of genus 2 curves. The other uses deformation theory, especially variations of Hodge structures of smooth hypersurfaces. (2) We also give an application to the zeta function of the curve X(u1, u2, u3, u4) when (u1, u2, u3, u4) ∈ Q4. We calculate an example that shows that the corresponding representation of Gal(Q/Q) is of GL2-type, as is expected for curves with real multiplications by cubic number fields.
","abstract_html":"<p>This thesis is devoted to proving the following:</p> <p>For all (u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>) in a Zariski dense open subset of <strong>C</strong><sup>4</sup> there is a genus 3 curve X(u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>) with the following properties:</p> <p>1. X(u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>) is not hyperelliptic.<br> 2. End(Jac((X(u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>))) ⊗<strong>Q</strong> contains the real cubic field <strong>Q</strong>(ζ<sub>7</sub>+ζ<sub>7</sub><sup>-1</sup>) where ζ<sub>7</sub> is a primitive 7th root of unity.<br> 3. These curves X(u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>) define a three-dimensional subvariety of the moduli space of genus 3 curves M<sub>3</sub>.<br> 4. The curve X(u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>) is defined over the field <strong>Q</strong>(u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>), and the endomorphisms are defined over <strong>Q</strong>(ζ<sub>7</sub>, u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>).</p> <p>This theorem is a joint result of J. W. Hoffman, Ryotaro Okazaki,Yukiko Sakai, Haohao Wang and Zhibin Liang. My contribution to this project is the following: (1) Verification of property 3 above. This is accomplished in two ways. One utilizes the Igusa invariants of genus 2 curves. The other uses deformation theory, especially variations of Hodge structures of smooth hypersurfaces. (2) We also give an application to the zeta function of the curve X(u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>) when (u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, u<sub>4</sub>) ∈ <strong>Q</strong><sup>4</sup>. We calculate an example that shows that the corresponding representation of Gal(<span style="text-decoration: overline"><strong>Q</strong></span>/<strong>Q</strong>) is of GL<sub>2</sub>-type, as is expected for curves with real multiplications by cubic number fields.</p>","abstract_has_math":false,"creators":["Liang, Dun"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-01-01T08:00:00Z","date_published":"2014-01-01T08:00:00Z","updated_at":"2026-07-24T02:58:17Z","subjects":["curves","jacobian","endomorphisms"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-11052014-135432","https://repository.lsu.edu/gradschool_dissertations/719"],"render_values":[{"text":"etd-11052014-135432","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/719","href":"https://repository.lsu.edu/gradschool_dissertations/719","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.719","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Liang, Dun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-10-20"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:09:45Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["curves","jacobian","endomorphisms"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-11052014-135432","10.31390/gradschool_dissertations.719","https://repository.lsu.edu/gradschool_dissertations/719"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is devoted to proving the following:
For all (u1, u2, u3, u4) in a Zariski dense open subset of C4 there is a genus 3 curve X(u1, u2, u3, u4) with the following properties:
1. X(u1, u2, u3, u4) is not hyperelliptic.
2. End(Jac((X(u1, u2, u3, u4))) ⊗Q contains the real cubic field Q(ζ7+ζ7-1) where ζ7 is a primitive 7th root of unity.
3. These curves X(u1, u2, u3, u4) define a three-dimensional subvariety of the moduli space of genus 3 curves M3.
4. The curve X(u1, u2, u3, u4) is defined over the field Q(u1, u2, u3, u4), and the endomorphisms are defined over Q(ζ7, u1, u2, u3, u4).
This theorem is a joint result of J. W. Hoffman, Ryotaro Okazaki,Yukiko Sakai, Haohao Wang and Zhibin Liang. My contribution to this project is the following: (1) Verification of property 3 above. This is accomplished in two ways. One utilizes the Igusa invariants of genus 2 curves. The other uses deformation theory, especially variations of Hodge structures of smooth hypersurfaces. (2) We also give an application to the zeta function of the curve X(u1, u2, u3, u4) when (u1, u2, u3, u4) ∈ Q4. We calculate an example that shows that the corresponding representation of Gal(Q/Q) is of GL2-type, as is expected for curves with real multiplications by cubic number fields.
"]},{"key":"dc:title","label":"Title","values":["Explicit Equations of Non-Hyperelliptic Genus 3 Curves with Real Multiplication by Q(ζ7+ζ7-1)"]}]}],"canonical_facts":{"dc:creator":["Liang, Dun"],"dc:date":["2014-10-20"],"dc:date.available":["2022-05-12T23:09:45Z"],"dc:description.abstract":["This thesis is devoted to proving the following:
For all (u1, u2, u3, u4) in a Zariski dense open subset of C4 there is a genus 3 curve X(u1, u2, u3, u4) with the following properties:
1. X(u1, u2, u3, u4) is not hyperelliptic.
2. End(Jac((X(u1, u2, u3, u4))) ⊗Q contains the real cubic field Q(ζ7+ζ7-1) where ζ7 is a primitive 7th root of unity.
3. These curves X(u1, u2, u3, u4) define a three-dimensional subvariety of the moduli space of genus 3 curves M3.
4. The curve X(u1, u2, u3, u4) is defined over the field Q(u1, u2, u3, u4), and the endomorphisms are defined over Q(ζ7, u1, u2, u3, u4).
This theorem is a joint result of J. W. Hoffman, Ryotaro Okazaki,Yukiko Sakai, Haohao Wang and Zhibin Liang. My contribution to this project is the following: (1) Verification of property 3 above. This is accomplished in two ways. One utilizes the Igusa invariants of genus 2 curves. The other uses deformation theory, especially variations of Hodge structures of smooth hypersurfaces. (2) We also give an application to the zeta function of the curve X(u1, u2, u3, u4) when (u1, u2, u3, u4) ∈ Q4. We calculate an example that shows that the corresponding representation of Gal(Q/Q) is of GL2-type, as is expected for curves with real multiplications by cubic number fields.
"],"dc:identifier":["etd-11052014-135432","10.31390/gradschool_dissertations.719","https://repository.lsu.edu/gradschool_dissertations/719"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["curves","jacobian","endomorphisms"],"dc:title":["Explicit Equations of Non-Hyperelliptic Genus 3 Curves with Real Multiplication by Q(ζ7+ζ7-1)"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:58:17Z"}